625 research outputs found
Large traveling capillary-gravity waves for Darcy flow
We study capillary-gravity and capillary surface waves for fluid flows
governed by Darcy's law. This includes flows in vertical Hele-Shaw cells and in
porous media (the one-phase Muskat problem) with finite or infinite depth. The
free boundary is acted upon by an external pressure posited to be in traveling
wave form with an arbitrary periodic profile and an amplitude parameter. For
any given wave speed, we first prove that there exists a unique local curve of
small periodic traveling waves corresponding to small values of the parameter.
Then we prove that as the parameter increases but could possibly be bounded,
the curve belongs to a connected set of traveling waves. The set
contains traveling waves that either have arbitrarily large
gradients or are arbitrarily close to the rigid bottom in the finite depth
case. To the best of our knowledge, this is the first construction of large
traveling surface waves for a viscous free boundary problem.Comment: 25 page
Traveling wave solutions to the one-phase Muskat problem: existence and stability
We study the Muskat problem for one fluid in arbitrary dimension, bounded
below by a flat bed and above by a free boundary given as a graph. In addition
to a fixed uniform gravitational field, the fluid is acted upon by a generic
force field in the bulk and an external pressure on the free boundary, both of
which are posited to be in traveling wave form. We prove that for sufficiently
small force and pressure data in Sobolev spaces, there exists a locally unique
traveling wave solution in Sobolev-type spaces. The free boundary of the
traveling wave solutions is either periodic or asymptotically flat at spatial
infinity. Moreover, we prove that small periodic traveling wave solutions
induced by external pressure only are asymptotically stable. These results
provide the first class of nontrivial stable solutions for the problem.Comment: Funding source for H. Q. N. correcte
Slowly traveling gravity waves for Darcy flow: existence and stability of large waves
We study surface gravity waves for viscous fluid flows governed by Darcy's
law. The free boundary is acted upon by an external pressure posited to be in
traveling wave form with a periodic profile. It has been proven that for any
given speed, small external pressures generate small periodic traveling waves
that are asymptotically stable. In this work, we construct a class of slowly
traveling waves that are of arbitrary size and asymptotically stable. Our
results are valid in all dimensions and for both the finite and infinite depth
cases.Comment: 22 page
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